Streaming currents in a single nanofluidic channel

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Streaming Currents in a Single Nanofluidic Channel

Frank H. J. van der Heyden, Derek Stein, and Cees Dekker

Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

(Received 4 May 2005; published 8 September 2005)

We report measurements of the streaming current, an electrical current generated by a pressure-driven liquid flow, in individual rectangular silica nanochannels down to 70 nm in height. The streaming current is observed to be proportional to the pressure gradient and increases with the channel height. As a function of salt concentration, it is approximately constant below 10 mM, whereas it strongly decreases at higher salt. Changing the sign of the surface charge is found to reverse the streaming current. The data are best modeled using a nonlinear Poisson-Boltzmann theory that includes the salt-dependent hydration state of the silica surface.

DOI:10.1103/PhysRevLett.95.116104 PACS numbers: 82.65.+r, 66.90.+r

Nanofluidics is an emerging field of interest because of its potential to study and manipulate fluids, nanoparticles, and molecules in lab-on-a-chip devices [1–5]. Electro-kinetic phenomena such as electro-osmotic flow and streaming currents [6 –8], known to be induced by the charged walls of a microchannel, will become increasingly important in the nano regime because of the larger surface to volume ratio [9]. Moreover, new physical effects may occur as the counterion distributions of electrical double layers at opposite surfaces of a nanochannel start to overlap [10]. So far, the effects of surface charge and potential on electrokinetic fluid phenomena in confined nanostructures have been largely unexplored.

Streaming currents provide a useful technique for study-ing the electric properties of the solid-liquid interface as they result from the transport of counterions in the electric double layer by an applied pressure gradient [Fig. 1(d)]. It has also been suggested that streaming currents in fluidic channels may provide a simple and effective means of converting hydrostatic pressure differences into electrical energy [11], and that this process becomes particularly efficient in the nano regime where double layers overlap [10]. Streaming currents have been primarily studied on systems involving many channels in parallel, using mate-rials such as sandstone cores [12], porous glass [11], and columns packed with latex beads [13]. While studies have appeared on single channels in the micron range [6,14,15], nanofabrication now permits us to investigate streaming currents in well-defined, nanoscale geometries where double layers may overlap.

In this Letter, we report measurements of streaming currents in individual rectangular silica nanochannels as a function of pressure, channel height, and salt concentra-tion. The data are compared with different models for the electrostatic properties of the surface: constant charge, constant potential, and a model that incorporates the chemical nature of the silica surface.

Nanofluidic channels were fabricated following a sili-cate bonding procedure similar to Stein et al. [9]. Channels 50 m wide and 4.5 mm long were patterned between

1:5 1:5 mm2 _{reservoirs on a fused silica substrate}

[Fig. 1(a)] by electron-beam lithography. The channels were plasma etched to a depth, h, ranging from 1147 nm down to 70 nm. After a thorough cleaning procedure that ended with a 1 min dip in 0.5% aqueous HF solution, the channels were sealed by first spinning a 20 nm layer of sodium silicate from 2% aqueous solution onto a flat fused silica chip and then pressing the silicate-coated surface to the patterned channel surface, and finally curing the device at 100C for 2 h. The channels were connected to a Plexiglas holder using silicone O rings to make a water-tight seal. Aqueous solutions, when inserted on one side of the holder, were drawn by capillary force into the nano-channels. Peek tubing connected one side of the channel to a pressurized chamber, while the opposite side was left open to atmosphere, as illustrated in Fig. 1(b). Electrical connections were established by inserting a Ag=AgCl

30 20 10 0 Current (pA) 4 3 2 1 0 Pressure (bar) 17µM 0.33M Pressure-driven flow (a) (c) (b) (d) counter-ions 4.5mm ~100nm high P air I 50µm P-sensor

FIG. 1. (a) Nanochannels are fabricated by bonding two pieces of fused silica. The top piece contains an etched nanochannel and two 1 mm diameter holes for fluid connections. (b) Side view of the nanochannel. (c) Streaming current as a function of pressure for a 140 nm high channel at low- and high-salt concentrations. Lines are linear fits. (d) Schematic illustration of the origin of the streaming current.

PRL 95, 116104 (2005) P H Y S I C A L R E V I E W L E T T E R S 9 SEPTEMBER 2005week ending

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electrode through a T junction on the pressurized side, and a plain Ag=AgCl electrode wire into the liquid on the opposite side. The applied pressure was controlled to val-ues of up to 4 bar using a Festo-LRP-1/4-4 pressure regu-lator and was continuously monitored using a calibrated Honeywell 24PC pressure sensor. A correction was made for the small pressure drop of about 2% across the channel entrance and exit, which was simulated using fluid dynam-ics modeling software. Potassium chloride (KCl) solutions ranging from 5:6 M to 1M were made by the serial dilution of a 1M KCl, 10 mM TRIS solution (pH 8:0) with distilled deionized water (18 M cm). A pressure of 4 bar was applied to remove any air bubbles from the channel and to achieve a stable streaming current. Approximately 5 h of stabilization time were allowed after initially filling the channel, and about 1 h for each subse-quent exchange of salt solution. After stabilizing, stream-ing currents were measured while applystream-ing pressures that were varied from 1 to 4 bar in 80 s steps.

The dependence of the measured streaming current, Istr,

on the applied pressure across a 140 nm channel is plotted in Fig. 1(c) for salt concentrations of 17 M and 0:33M. Clearly Istrscales linearly with pressure for both salt

con-centrations. We define a streaming conductance, Sstr, as the

streaming current per unit applied pressure. Sstr is clearly

higher for the low KCl concentration, as can be seen from the slopes in Fig. 1(c). This contrasts the electrical con-ductance in nanochannels, which decreases at low salt [9]. Figure 2 shows S_stras a function of KCl concentration, c, for a 140 nm channel. Data points are averaged over two data sets, and the error bars indicate the average standard deviation for all points. The data show that S_stris approxi-mately constant for c below 10 mM, and decreases at higher c where it has dropped by an order of magnitude at c 1M KCl. The same trend is observed for different channel heights as shown in Fig. 3, which shows Sstrversus

c for channels ranging from 70 to 1147 nm. The open symbols represent data points at very high molarity, where values for S_strappear to be depressed. We attribute this to deviations from the mean-field model or to effects of surface roughness, as the double layer is very compact here. Higher channels are found to yield a larger streaming current. For all channel heights, Sstr reaches a constant

value in the low-salt regime, and decreases at high salt. The low-salt plateau extends to higher c for smaller channels.

We begin modeling these data by equating the streaming current I_str to the product of the charge density, x, and the local speed of the fluid, ux, integrated over the cross section of the channel:

Istr w

Zh=2 h=2

xuxdx; (1)

where x is the height from the channel midplane, w is the channel width, and h is the channel height. We ignore sidewall effects because w h, and treat the flow and charge density profiles as uniform over the width of the channel, as in an infinite parallel-plate geometry. The velocity distribution is described by a Poiseuille flow sub-ject to the no-slip boundary condition at the walls: ux P

8Lh

2_4x2_{, where L is the length of the channel, P}

the pressure difference, and the viscosity. We describe

x using nonlinear Poisson-Boltzmann theory, starting with the Poisson-Boltzmann equation for the electrostatic potential distribution x: d2_x dx2 kT2 e sinh _{e x} kT ; (2)

FIG. 2. Streaming conductance as a function of KCl concen-tration for a 140 nm high channel. Lines show model curves of constant surface charge, constant zeta potential, and the chemi-cal equilibrium model, treated here in this Letter.

FIG. 3 (color online). Streaming conductance as a function of KCl concentration for five different channel heights. Data points are averages of two data sets. The dashed lines represent the chemical equilibrium model for a single set of parameters, whereas the solid lines represent the same model when the parameters C and pK are different for each curve.

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where e is the electron charge, k is the Boltzmann constant,

T is the temperature, and 1 is the Debye screening length, defined by 2 2e2_n

0kT. Here, 0 is the permittivity

of the vacuum, is the dielectric constant of water, and n is the number density of the monovalent ions, proportional to

c. The analytical solution for x is given by [16]

x ₀ 2kT

e lnJacCDzjm; (3)

where ₀ is the potential in the center of the channel, and JacCDzjm is the Jacobi elliptical function with argument

z x 2 e

e 0=2kT _{and parameter m e}2e 0=kT_{. The}

elec-tric field, Ex, and the charge density, x, are obtained from x through Ex d x_dx and Poisson’s equation,

x 0d

2_x

dx2 .

We now compare the predictions of this model for the streaming current, assuming different boundary conditions for the electrical surface properties: constant surface charge , constant zeta potential , or a chemically vari-able . Constant is imposed using Gauss’ law:  ₀Ex h

2, and constant by specifying the

poten-tial at the no-slip plane: x h 2.

For any given h and , Eq. (3) is solved using either of these boundary conditions to give a value for 0, and

thereby x and x are solved through Eq. (3) and Poisson’s equation. In the third model, we account for the chemical reactivity of the surface by allowing to vary due to proton transfer through the following equilib-rium [17]:

SiOH SiO H

: (4)

Behrens and Grier [16] developed a model for the diffuse layer potential, which we take to be , by describing the dissociation of SiOH groups using equilibrium constant,

pK, and the compact Stern layer using capacitance, C,

yielding the following relation for :

 kT e ln e  ln10 e=kT pK pH  C; (5)

where is the surface density of chargeable sites. A second relation between and derives from the charge and potential distribution in the diffuse layer. For the case where h  1, no double layer overlap occurs, and the surface charge density is approximated using the Grahame equation:  20kT e sinh _e 2kT : (6)

Solving Eqs. (5) and (6) yields a value for the surface charge , or potential . When double layer overlap does occur, no analytical solution is available. In this case x, and thereby , were solved for a set of values as a boundary condition. was plotted as a function of , and its intersection point with the chemical equilibrium

relation, Eq. (5), was determined. The intersection point yields self-consistent solutions for and that both de-pend on and h. The thus obtained was used as a boundary condition to find x and S_str. The model pre-dicts that increases with the salt concentration and that its dependence on h is relatively unimportant for the chan-nel sizes and salt concentrations that we have used: at the lowest salt concentration, 5:6 M, and the smallest chan-nel, 70 nm, is predicted to be reduced by just 1.3% relative to the largest channel of 1147 nm.

We now compare these three models for the streaming current in silica nanochannels. The predictions for S_strin a nanochannel as a function of c are plotted in Fig. 2 for  60 mV, 4 mC=m2_{, and the chemical equilibrium}

boundary conditions with parameters 8 nm2, pK 7:9, pH 8, and C 0:3 Fm2. Note that the latter val-ues are not free fit parameters, but were adopted from literature [18]. From Fig. 2, it is clear that the constant- line fails to describe our data, as it is constant at high salt and drops when c is decreased below 100 M. The constant- model, which previously described the salt dependence of the electrical conductance successfully [9], does a somewhat better job: it yields a plateau at low salt and drops off at high salt. However, the current de-crease sets in at significantly lower c (1 mM) than ob-served (10 mM), leading to significant deviations be-tween model and data. The chemical equilibrium model gives the best fit over the full range of c, predicting a roughly constant low-salt plateau, and the correct high-salt decay.

In Fig. 3, the dashed lines represent predictions of the chemical equilibrium model for the same single set of parameters that are applied to all curves [18]. This model describes the data reasonably well over 5 orders of magni-tude in salt concentration and a factor of 16 in channel height, using a single set of material parameters. Since the properties of silica can vary between channels, we may fit the parameters pK and C for each curve [19] to obtain a very good description of S_str for all channels, presented using solid lines.

We also demonstrate the sensitivity of the streaming current in nanochannels to the sign of the surface charge by inverting the surface charge. For this purpose, we coated a fused silica nanochannel with a layer of polylysine by incubating the channel in 0.01% 70 –150 kDa polylysine solution for 12 h, which is expected to render the surface positively charged. Indeed, Fig. 4 shows that I_str in the direction of the liquid flow is negative for polylysine-coated channels, which indicates that the double layers are populated with negative ions, whereas the untreated silica surface generates a positive current, indicating posi-tive ions in the double layer [20].

Finally, we comment on the proposal that the streaming-current effect can be used to convert hydrostatic energy into electrical energy [11]. Using the exact Poisson-PRL 95, 116104 (2005) P H Y S I C A L R E V I E W L E T T E R S 9 SEPTEMBER 2005week ending

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Boltzmann solution for the charge distribution, combined with the chemical equilibrium boundary condition for the silica surface using a single set of material parameters [18], we find that a maximum energy conversion efficiency is obtained in the regime of double layer overlap. In aqueous solutions, this limits optimal channel dimensions to the submicron range. For example, for a 105MKCl aqueous solution a maximum energy conversion efficiency of 6% is reached in a channel of 145 nm height, which is well within our experimental range. Details will be published elsewhere.

In conclusion, we have studied streaming currents in individual silica nanochannels with heights as small as 70 nm and found that they increase linearly with applied pressure. The streaming current reverses direction when polylysine-coated, positively charged channel walls are used instead of negatively charged fused silica, empha-sizing its sensitivity to the polarity of the surface charge. The streaming current increases with channel height, is approximately constant at salt concentrations below 10 mM KCl, and drops at higher salt. It can be modeled using nonlinear Poisson-Boltzmann theory, combined with a pressure-driven Poiseuille flow in a parallel-plate geome-try. A chemical equilibrium model for the surface charge, taken as a boundary condition, predicts streaming currents in nanochannels fairly well over the full range of measured heights and KCl concentrations, whereas constant surface charge, and in particular constant potential boundary con-ditions, gave a poor description. This highlights the im-portance of using appropriate boundary conditions when dealing with electrokinetic phenomena in nanofluidics.

The authors acknowledge discussions with Serge Lemay, and funding from NWO and FOM.

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[18] Reported values for the parameters in the chemical equi-librium model have ranges for 5–8 nm2_{, pK}

6–8:5, and C 0:2–2:9 Fm2 [R. E. G. van Hal et al., Adv. Colloid Interface Sci. 69, 31 (1996); S. Ong et al., Chem. Phys. Lett. 191, 327 (1992); and Refs. [16,17] ].  is weakly sensitive to , and increases with C and pH 

pK. For pH pK between 0:5 and 1:5 and C between

0.2 and 2:9 F=m2_{, we find extremes for at low salt}

between 2 and 8 mC=m2_{, and at high salt between}

15 and 192 mC=m2_{. In all calculations we used values}

of 0:8904 mPas and 78:54 for water at 25_{C, at}

which temperature our experiments were performed. Parameters 8 nm2, pK 7:9, pH 8, and C 0:3 Fm2 predict the data best for all channel heights within the reported parameter space.

[19] Parameter values for the solid lines in Fig. 3: h 70 nm ! pK 6:8, C 0:26 Fm2; h 140 nm ! pK 7:8, C 0:26 Fm2; h 279 nm ! pK 8, C 0:2 Fm2; h 563 and 1147 nm ! pK 9, C 0:9 Fm2_{. 8 nm}2 _{and pH 8 are kept constant.}

[20] At low salt we observe that the streaming current of the polylysine-coated channel scales nonlinearly with the pressure. We speculate that long polymer chains at the wall may swell at low salt, thereby reducing the effective height, whereas at higher pressure, shear forces at the wall straighten out the polymers by fluid drag, thus increasing the effective height and streaming current.

FIG. 4. Effect of the sign of the surface charge on the stream-ing current. The streamstream-ing current is shown as a function of pressure for a 279 nm high nanochannel at low- and high-salt concentration. Data are shown for a negatively charged silica surface, and for a polylysine-coated surface that is positively charged. Lines indicate linear fits.